29-12-2012, 06:05 PM
Critical stress and load conditions for pitting calculations of involute spur and helical gear teeth
Critical stress and load conditions for pitting calculations.pdf (Size: 721.56 KB / Downloads: 109)
Introduction
To compute the pitting load capacity of spur and helical involute gears, the most widely used standards as AGMA [1,2] or ISO [3]
employ the Hertz equation to evaluate the contact pressure. The load is assumed to be uniformly distributed along the line of
contact, though it is known that the load distribution depends on the meshing stiffness of the pair of teeth, which is different at any
contact point. This means that the load per unit of length is also different at any point of the line of contact, and therefore several
influence factors for load distribution are required to correct the calculated values of the bending and contact stresses [1,4].
Some studies on the load distribution along the line of contact can be found in technical literature [5–13], but all of them
provide results obtained by numerical techniques or the finite element method (FEM). Those methods allow to obtain some
conclusions regarding the considered gear pair, but make it very difficult to extract general conclusions, valid for any gear pair.
In previous works [14–19], the authors obtained a new load distribution model from the minimum elastic potential energy
criterion. The elastic potential energy of a pair of teeth was calculated and expressed as a function of the contact point and the
normal load. The load sharing among several pairs of contacting teeth in spur gears was obtained by solving the variational
problem of minimizing the total potential energy (equal to the addition of the potential energy of each pair at its respective contact
point) taking into account the restriction of the total load to be equal to the sum of the load at each pair. The same approach was
used for helical gear teeth by dividing each helical pair in infinite slices, perpendicular to the gear axis, assuming each slice to be
equivalent to a spur gear with differential face width, and extending the integrals to the complete line of contact. This approach
allowed the value of the load per unit of length at any point of the line of contact and at any position of the meshing cycle to be
known. Through this model, some preliminary studies on the location of the points of critical stress and the determinant load
Load distribution model
Reference [20] presents in detail the model of load distribution of minimum elastic potential energy. In general terms, that
model was obtained by computing the total elastic potential energy from the equations of the theory of elasticity, considering all
pairs of teeth in simultaneous contact, with an unknown fraction of the load acting on each one, and minimizing its value by means
of variational techniques (Lagrange's method). It has been demonstrated that the load per unit of length depends on the inverse
unitary potential v(ξ), which is defined as the inverse of the elastic potential for unitary load and face width. Obviously, the inverse
unitary potential depends on the contact point, which is described by the ξ parameter of the contact point at the pinion profile
as:
Accuracy of the method
The previous section stated that Eq. (25) is accurate enough for evaluating the maximum of functionΦ(ξ), as well as Eq. (29) to
evaluate the maximum of function Φ(ξ)/Iv(ξ0) for the cases in which simultaneity is not given. This section presents a complete
study of the accuracy of both methods.